Note, in view of Corollary REF (b), we have \(\widehat{\operatorname{Tor}}_{i}^R(M,N)=0\)  for all \(i\in \mathbb {Z}\) . Therefore, \(\operatorname{Tor}_i^R(M,N)=0\)  for all \(i\gg 0\) . Hence, by [1]}, it follows that either \(\operatorname{pd}(M)<\infty \)  or \(\operatorname{pd}(N)<\infty \) , i.e., either \(M\)  or \(N\)  is free.
